3 Lattices for products of trees

Let $T_1$ and $T_2$ be locally finite trees. Then the group $\Aut(T_1 \times T_2)$ is isomorphic to $\Aut(T_1) \times \Aut(T_2)$. A lattice $\Gamma \leq \Aut(T_1 \times T_2)$ is irreducible if it is not commensurable to a product $\Gamma_1 \times \Gamma_2$ of lattices $\Gamma_i \leq \Aut(T_i)$ for $i = 1,2$.

Problem 3.1 (Comments) [Burger--Mozes] Let $\Gamma$ be an irreducible uniform lattice in $\Aut(T_1) \times\Aut(T_2)$. For $i = 1,2$, let $H_i$ be the closure of the projection $\pi_i:\Gamma \to \Aut(T_i)$. What are the possible $H_i$s? What about nonuniform lattices?

Remark We assume the projections are locally primitive, that is, that the restriction of their action to the star of any vertex of the tree $T_i$ is primitive. If the irreducible lattice $\Gamma$ is uniform then the projections cannot be dense in either of $\Aut(T_i)$ (Burger--Mozes MR1839489). If $\Gamma$ has an infinite linear representation over characteristic $0$, then $\G$ is an extension of an arithmetic lattice (Burger--Mozes--Zimmer BurgerMozesZimmer). D. Rattaggi's thesis Rattaggi gives many interesting examples.

Problem 3.2 (Comments) [Burger--Mozes] Study lattices in products of three or more trees.

Remark One would like to construct interesting lattices. Here "interesting" means, in particular, examples which are not arithmetic or which are not extensions of arithmetic lattices. It is not obvious how to make deformations with $\geq 3$ trees. Maybe these examples will give new phenomena e.g. a group which is residually finite and not linear?

Problem 3.3 (Comments) [Burger--Mozes] Define and study lattices in an infinite product of trees: "adeles".

Remark There is a chance that all lattices are arithmetic here.

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Lattices in Trees and Higher Dimensional Complexes/Section3 (last edited 2008-02-01 15:25:16 by RickScott)